Another guest post from Katherine Cook. Enjoy!

“Your brain manages a vast information highway–yet a simple math problem can create a traffic jam that brings everything to a halt.”

So begins an article on the The Brain by Carl Zimmer in the November 2010 issue of Discover Magazine. The Brain. The squishy, ponderous, specialized mass of flesh suspended in our craniums that gives us each our senses, all the way down to our sense of I. Without the brain, one could and probably does argue, there is no us. The brain is the thing that brings the massive, beautiful, incomprehensible universe we live in into some kind of focus, and yet it stalls out on simple math problems.

The example Carl starts us off with is the following: “Pop Quiz: What is 357 times 289?” He admonishes us to reject our pencils and calculators and nobly go it alone, and then waits impatiently through the next few sentences until his point emerges: this calculation is, for most us, impossibly slow. It’s not as though we can’t do it. After all, most of us remember an algorithm, some algorithm, from childhood, maybe the one where you multiply the 7 by the 9, then the 5 by the 9, then the 3 by the 9, then write down a placeholder zero in the next line for some reason none of us really can remember or perhaps never knew, then do the same only with the 8 in place of the 9, and so on, until we arrive at the finish line, huffing and puffing, but bravely asserting that 357 times 289 is 103,173. Of course, that’s if the algorithm yielded no errors, which given how many individual calculations were required to see it through the end, is not guaranteed by any means. Anyway, the point is that most of us could, hypothetically, crunch the calculation. His point, Carl Zimmer’s, is it’s a curious thing indeed that the human brain wields massive computational power and yet hiccups over the smallest, the piddliest, the most trivial of arithmetic calculations. Really, a multiplication problem of this level takes, as he says, a ‘puny amount of processing’. Compare that to the brain’s ability to recognize faces, to identify certain kinds of patterns, to regulate homeostasis through the body and manage complex motor skills at the same time, and one is left with a question. What’s with this processing discrepancy?

Of course, there is some subtle kind of a problem in the comparison. It often seems that much of what we do in terms identification of faces, for example, involves ball-parking a solution. We estimate the rough sketch of a face we know and use that to identify a familiar person. This is why we’ve all had the experience of mistaking a stranger for someone we know. Our brains aren’t equipped to produce an exact replica of a face we know: if you close your eyes right now and try to describe the face of a good friend, you’ll probably trip up on the specifics. For many of us, when we subconsciously sort the world into categories friend and foe, known and unknown, it seems that we’re using fuzzy edges to define our containments. Generally that’s good enough.

In math, we’re taught not to do this. ‘Use the algorithm!’ we are enjoined, and from very early on in our lives. We are so caught up the in the algorithm for processing the number that we don’t even know what we are doing. Maybe you worked through the calculation above in your head and you kept getting lost, forgetting whether you were multiplying 9 times 5 or 8 times 5, or maybe you lost some number you were carrying somewhere else. Maybe, heaven forbid, you forgot the placeholder zeroes and couldn’t figure out why your answer was so small (and maybe you didn’t even notice in that case that it was much smaller than it should be). If you were having a moment in the zone, maybe you asked yourself what all the carrying is supposed to be about anyway, and maybe, if you aren’t ruined by all that enjoining done in grade school, you remember that the whole point of this exercise was to combine groups of 357 a total of 289 times. Wait a minute. Why are we forcing our minds into manipulating a level of detail that is not natural for us? Why not estimate this product, just like we do everything else? 289 is awfully close to 290, likewise for 357 and 360. Even if we round 289 to 300, we get that the product is  108,000, which is close. But we overestimated 289 by about 10, and 10 groups of 360 is 3600, so we overestimated about that much. That get us to 104,400. Within spitting range on a number so large. For the average person, this is within the ‘excuse me, I mistook you for someone else’ range.

None of this is Carl Zimmer’s point. His point is that there are many steps involved in solving this problem. Enough to cause a mental traffic jam as all the little products collide in your head with the carried ones, the casualty placeholder zeroes trotted off scene on a stretcher. This is due to what psychologists term a ‘psychological refractory period’, a waiting time in between pulses of thought during which the mind essentially resets itself. As we churn through the long mental multiplication, each step has to wait for the first to be completed and processed before it can be moved through the system. Psychologists have done a number of experiments to demonstrate the mental bottleneck that occurs when processing certain kinds of information, and the research seems to suggest the bottleneck occurs most when the same part of the brain is called upon repeatedly to process the same kind of information. For most of us there is no bottleneck when we drink a cup of coffee, read an article, and listen to music at the same time, but it shows up when trying to solve a long multiplication problem in our heads. We solve the first simple step in the algorithm, and then there is a refractory period before we move onto the next.

Even if a person abandons the algorithm and goes with more rapid estimations (and it would be interesting to see experiments on the change in refractory period when the brain is more engaged with the larger picture of the multiplication problem, rather than being numbed out by the nonsensical algorithm), sometimes a multiplication problem is just a multiplication problem. They can be slow and non-elucidating of the world of mathematics underneath. Many people know math simply on the level of computation, and if every time they do some kind calculation the entire complex superhighway of their mind shuts down for the duration of the tedious, opaque, and frustrating calculation, no wonder so many people look down on the subject. Who among us enjoys feeling such a total power-down? But when a person is doing real and deep mathematics, they are engaged on many levels; creatively, intuitively, analytically. The secret truth about mathematics is that with the deep stuff, there is no bottleneck. Focus, yes. Bottleneck, no. The real problems of mathematics call upon different areas of the brain and they all come on-line and fire in concert. Like all creative pursuits, there are blocks and obstacles to be overcome, but these aren’t the bottleneck of Carl Zimmer’s article either. The bottlenecks of the mental traffic jam sort are the result of repeatedly calling on one area of the brain to perform: calculate! again! again! again! But when a mathematician is humming along on a complex and mysterious problem, the brain is operating more like it was built to; different areas working in conjunction to understand the problem not just computationally, but spatially, in terms of patterns, calling upon her memory of previous problems, simplifying, and finding any cracks in the wall to which she can apply her ferocious might. Though she may be gloriously stuck, her mind is not shut down to a single-file line of boring calculations. Stuck is not bottlenecking. Stuck is reassessing the landscape with all the tools you have at your disposal. The algorithms so many people identify as the sum of all things mathematical are disappointing to everyone, including mathematicians. But thankfully, most of us don’t really do them. And we just look confused when we are told yet again that someone hates math. Maybe this will start to explain a bit of the difference.

Comments 6

  1. Paul Hawking

    This is a brilliant synthesis of the original article combined with an insightful take on why mathematics may be more of a mental overload than a “smartness” limitation for many students. It is ironic that the focus of many remedial math courses is on teaching tricks for remembering algorithms when more time spent on projects and investigations, where genuine creativity can play a part in the learning, could prove more fruitful.

    1. Katherine

      I like your point that maybe the kids who are left behind on these facts are dealing with a more limiting bottleneck, rather than a deficiency of intelligence. If only students were given more options to explore a vision of mathematics that calls on different parts of the mind. Anecdotally, I can speak to the experience of watching a third grader enter a brain tunnel when they are trying to recall multiplication facts in their head, and yet employ some pretty sophisticated problem solving strategies to, say, count how many squares there are on a chessboard. It is unfortunate that so many never get try something like the latter and identify math with the former.

  2. Kalid

    Great post. This reminds me of the idea of emulation/virtualization in computer science. Basically, you have hardware (the computer chip) and software (instructions to run). One interesting thing you can do is have your software simulate another piece of hardware, and then run software inside this second level.

    It’s used often times in video game systems (Playstation 3 can run Playstation 1 games — the PS3 simulates the earlier version in software) and on the desktop with virtual machines (I can run Windows XP within MacOS by creating a “Virtual Machine”).

    Unfortunately, the main drawback is that there’s a huge performance cost when running this way, vs. running software meant for the native hardware. In a way, part of math is getting us to think about things natively, instead of trying to simulate an abacus inside our heads.

    1. Katherine

      It’s interesting to think of the comparison with hardware and software. Processing power was one of the points in the original article: mainly, that the human brain outperforms and outprocesses the most sophisticated computers, except for on some of these particular types of problems (i.e. multiplication). It seems that for much of our most efficient and effective processing, we are being creative in conjunction with recall and crunching data, such as when we remember things from the past (research indicates we recreate memories rather than just pulling a file). It might be my bias, but I suspect that our native hardware is built to create throughout everything we do. Or at least most of the fun stuff.

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  4. Tristram Bogart

    I love this post, especially your point that many of the things that humans effortlessly outperform computers at, such as face recognition, involve estimation and patterns rather than exact memory/calculations. But oddly, I don’t think Zimmer’s example of multiplying three-digit numbers fits his own main point! If the difficulty in multiplying the numbers were mainly that of repeating the same type of operation many times, then (as long as we use the same algorithm) it would carry over to doing the multiplication on paper, which Zimmer admits is much easier.

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